# Teaching and learning of mathematics

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An Overview

This research paper is to discuss about the nature and history of mathematics, how it has taken its place within the National Curriculum; the framework for teaching Mathematics in Secondary and finally will be how the subject has been taught and learned in secondary schools.

Introduction

Education has made a difference in my life, the knowledge I have gained has given me the potential to explore, think and make decisions accordingly. In other words, Education is a powerful tool and plays a vital role to shape up a strong economy of a country.

As a Mathematics teacher, I clearly understand my key role in imparting knowledge and skills to the younger generation to make full use of their potential.

The perception of mathematics has been changed over the years. Hence, it is important to look back at the nature of mathematics, how it has taken its place within the national curriculum; how the teaching and learning of mathematics has been developed over the years and finally as a secondary school teacher I will be exploring the current National Strategies (Secondary) for mathematics to see how I can apply the framework on my teaching practice in near future.

Nature of Mathematics

Even though mathematics is one of the many subjects in schools, there is a greater pressure on pupils to succeed in Mathematics other than subjects like History, Geography; why is that so?

As part of my investigation into the nature of Mathematics I referred to two sources that gave substantial evidence towards the nature of Mathematics.

1) __The Enquiry Committee__: A Major Enquiry Committee was set up in 1978 to consider the teaching of Mathematics in Primary and Secondary schools. After 4 years of study and research the committee came out with a report called The Cockcroft Report.

'It would be very difficult - perhaps impossible - to live a normal life in very many parts of the world in the twentieth century without making use of mathematics of some kind.' (The Cockcroft Report (1982), Mathematics counts)

This fact itself for a thought is sufficient to reason out the purpose of importance given in teaching and learning mathematics in Schools.

The usefulness of Mathematics can be perceived in different ways; as arithmetic skills needed to use at Home and Office, as basis for development of Science and Technology and usage of Mathematical techniques as management tool in commerce and industry. Therefore, the Enquiry Committee in their report (The Cockcroft Report) concluded that all the perceptions on usefulness of mathematics arise from the fact that mathematics provides a mean of communication which is powerful, concise and unambiguous. Hence, providing a principal reason for teaching mathematics at all stages in the curriculum.

2) According to American Association for the Advancement of Science (AAAS), mathematics is closely related to Science, Technology and being greatly used in real life. The association has launched a program called Project 2061 where they relate mathematics into Science and Technology.

Project 2061 is an ongoing project that was launched in 1985 in America, where its main objective is to help all Americans to literate in Science, Mathematics and Technology. As part of the project, it has been clearly defined that mathematics does play an important role in developing Science and Technology in real life. Hence, it adds to the importance for teaching Mathematics in Schools.

Besides communication, Mathematics can be used to present information by using charts, graphs and diagrams. The following bar chart for Moscow, Russian Federation shows the years average weather condition readings covering rain, average maximum daily temperature and average minimum temperature. (http://www.bbc.co.uk/weather/world/city_guides)

Hence, this is a typical example of using Mathematics in real life context.

As what AAAS has mentioned about the Mathematical representation, manipulation and derivation of information based on a mathematical relationship formed; the enquiry committee as well does mention in its report the usage of figures and symbols in mathematics for manipulation and to deduce further information from the situation the mathematics relate to. They gave 3 scenarios;

A car that has travelled for 3 hours at an average speed of 20 miles per hour; we can deduce that it has covered a distance of 60 miles.

To find the cost of 20 articles each costing 3p, the area of carpet required to cover a corridor 20 metres long and 3 metres wide

In the 3 scenarios, we made use of the fact that: 20 x 3 = 60; hence it provides an illustration of the fact that the same mathematical statement can arise from and represent many different situations. This fact has important consequences. Because the same mathematical statement can relate to more than one situation, results which have been obtained in solving a problem arising from one situation can often be seen to apply to a different situation.

Thus this characteristic of Mathematics does show its importance in the study of science and Technology as covered in PROJECT 2061 by AAAS.

HISTORY OF MATHEMATICS

By looking at the history of Mathematics; it has been further proven how the development of mathematics had impact on development of Science and Technology.

There have been contributions from various cultures to the development of mathematics over the centuries;

Under the Egyptian Mathematics; the oldest mathematical text discovered so far is the Moscow papyrus. It consists of what are today called word problems or story problems, which were apparently intended as entertainment. The Rhind papyrus is another major Egyptian mathematical text, an instruction manual in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge including composite and prime numbers; it also shows how to solve first order linear equations as well as arithmetic and geometric series.

Greek mathematics was more sophisticated than the mathematics that had been developed by earlier cultures. Greek mathematics is thought to have begun with Thales (c. 624-c.546 BC) and Pythagoras (c. 582-c. 507 BC). Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. Pythagoras is credited with the first proof of the Pythagorean Theorem, though the statement of the theorem has a long history. Both Geometry and Pythagoras' Theorem are being taught as major topics in current curriculum.

The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, Johannes Kepler, a German, began to work with this data. In part because he wanted to help Kepler in his calculations, John Napier, in Scotland, was the first to investigate natural logarithms. Kepler succeeded in formulating mathematical laws of planetary motion. This explains the relationship between mathematics and science or another word, how knowledge of mathematics has been used to develop science over the years.

The most influential mathematician of the 1700s was arguably Leonhard Euler who popularized the use of the Greek letter p to stand for the ratio of a circle's circumference to its diameter.

The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, the British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1 and in which, famously, 1+1=1. Boolean algebra is the starting point of mathematical logic and has important applications in computer science.

Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.

The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics was awarded, and jobs are available in both teaching and industry.

Therefore, from the 20th Century is where importance has been given to teaching of mathematics.

FORMATION OF NATIONAL CURRICULUM

This further explains how the national curriculum for Mathematics has been formed in Britain. Let's look at the various views of Mathematics usage in Industry before the Enquiry Committee was set up;

From 1973 to 1976 there were a large volume of complaints which seemed to be coming from employers about lack of mathematical competence on the part of some school leavers; this was evident from the speech made by Mr James Callaghan, at that time Prime Minister. He raised the concerns about the standard of numeracy skills of school leavers. There were written evidence by the Confederation of British Industry (CBI) stating the concerns that employers are raising with regards to the insufficient arithmetic skills acquired by young people when they leave school at the statutory school. Thus, leaving an impression there are no proper coordination between the requirement of employers and what is delivered by schools. This was further proven by the training officers within the CBI industry where they witness the young people at the age of 16+ could not pass simple tests in mathematics and required remedial tuition before enrolling themselves into further education courses. In a written evident, the Engineering Industry Training Board does emphasis that Mathematics is about understanding of concepts and requires innovative way to teach it.

Therefore, these are the examples of complaints received and the main reason for the enquiry committee to set up in 1978 to investigate complaints about low levels of numeracy among young entrants to employment and the need for improved liaison between schools and industry. Hence we could deduce that due the mathematical knowledge demand in the work force has brought mathematics an important place in the national curriculum to promote numeracy skills among the young people.

MATHEMATICS IN NATIONAL CURRICULUM

To make up all the deficiencies that has been occurred in earlier years, the current National curriculum has set out guidelines for schools to design their curriculum for mathematics to meet the national standards.

As stated in the Programme of Study (POS) for mathematics, the curriculum aims to develop young people through mathematics as successful learners where they would enjoy learning, confident individuals and responsible citizens who will make positive contribution to the society. As a teacher of mathematics, I think it is essential for me to know the expectations of the curriculum for the young people in learning Mathematics as I could deliver my lessons accordingly.

As mentioned by the Engineering Industry Training Board, understanding the concepts in Mathematics is essential, hence, within the mathematics curriculum a set of key concepts outlined where pupils are expected to be competent in communicating and applying mathematics within and beyond classroom; to be creative in using mathematics knowledge to solve problems.

Bearing this in mind, next I am going to look into the key processes that have been set out for pupils to know in order to deepen and broaden their knowledge, skills and understanding of Mathematics.

The key processes of mathematics have been clearly outlined between key stage 3 and key stage 4. At these both stages pupils are expected to build on skills to represent, analyse, interpret, evaluate, communicate and reflect on mathematical problems. These skills are essential for pupils to progress within the subject and as well they could use these tools in understanding science, engineering, technology and economics. This further evident the importance of learning mathematics and as well teaching it in schools.

PERSONAL DEVELOPMENT (ECM) THROUGH MATHEMATICS

In building ECM outcomes into my lesson is essential as it is vital part of personal development for every pupil in school. Through my research and lesson observations I have realised through mathematics I could embed the five ECM outcomes into my lessons. Further more, the programme of study for mathematics provides opportunities to plan sequences of lessons to support that support personal development through the five ECM outcomes. They have stated that mathematics can be enjoyed from its creative and investigative aspects and pupils could achieve from developing mathematical ways for understanding the economy. Hence, pupils could develop their problem solving, decision making and reasoning skills through a range of tasks.

By understanding the numerical data, mathematics enables pupils to use the skills in making both healthy and financial decisions. At the same time, I could embed strategy games and logic puzzles into my lessons that promote maintaining mental health.

Pupils through mathematics learn to understand to use probability scale to estimate risk and this is a key factor for them to stay safe by making a balanced risk decisions.

The skills of reasoning with numbers, interpreting graphs and diagrams and communicating mathematical information are vital in enabling individuals to make sound economic decisions in their daily lives.

Having confidence and capability in mathematics allows pupils to develop their ability to contribute to arguments using logic, data and generalisations with increasing precision. By becoming skilled in mathematical reasoning it allows pupils to have a positive contribution to the development of the society.

Hence, again by relating ECM outcomes to Mathematics, it is clearly known the importance of learning this subject and the requirement of various approaches in teaching it.

Functional Skills in Mathematics

Functional mathematics provides pupils with the skills and abilities that they require to play a responsible role within their community and be able to operate confidently in life and solve problems from a wide range of contexts. The programme of study has embedded functional skills as a subset of Key processes to teach mathematics. Therefore the key processes for representing, analysing, interpreting, evaluating, communicating, and reflecting comprises the skills required for functional mathematics. Hence, as part of my research I have looked into case studies from the National Curriculum website (http://curriculum.qcda.gov.uk) where functional skills has been applied.

CASE STUDY USING FUNCTIONAL SKILLS

1) At Wellacre Technology and Vocational College, a year 9 science project on skiing was given to pupils to understand the relevance of mathematics in real life. The pupils were asked to calculate surface area of their feet and pressure as to ensure their skis did not sink into the snow. Hence, from this project by using a set of real figures proved to be incentive and challenging for the pupils as they need not have to go back to the traditional system of looking at text book answers.

By looking at this case study, it gives me an idea of how to implement functional skills into my lessons as to relate maths to every day life.

2) At Lancaster Girls' Grammar School, a year 10 group of pupils were introduced open - ended projects where they are required to use mathematics to solve real life problems.

In year 10 pupils were encouraged to make links between mathematics and music. Some considered what kinds of functions might be used to model sound waves. Others explored the connections between the fibonacci sequence and the layout of a keyboard.

In this project, pupils defined their own problem, decided on the data to collect and how to collect it, gathered information from a number of sources, including their parents or other pupils, considered how to analyse their data, used and applied mathematics skills and drew conclusions. At the end of the projects, they presented their findings and evaluated how successful they had been.

Therefore, Staff and pupils embraced the new way of working. The head of department acknowledged that 'it was a considerable risk to introduce this way of teaching but it paid off. Initially, staffs were concerned about setting problems when they did not know the answers but once the work was underway they enjoyed a different way of teaching. The projects offered opportunities to stretch pupils and encourage them to make connections between different parts of their learning.'

Again by relating mathematics to real life, it adds value to the learning of mathematics for pupils in schools.

ASSESSING PUPIL'S PROGRESS IN MATHEMATICS (APP)

In order to confirm that there is an effective learning is taking place based on the teaching in a classroom environment, it is important to assess the pupils' progress. Hence by assessing pupils' progress it gives evidence to learning and to restructure our teaching strategy based on the assessment made on pupils' progress in learning.

There are various ways to assess a pupil's learning (AFL) during a lesson. The most common and traditional way to assess for learning is of course marking of books and worksheets given during the lesson. By doing so, a teacher realises whether an individual pupil has understood a topic taught in the class and gives a guide for the teacher to plan the next lesson accordingly. With such assessment, it is possible to differentiate the various learning style of the pupils and again it allows the teacher to plan the lesson accordingly. Another words, by assessing the learning of pupils, a teacher is able to identify individual pupil's learning need.

Assessment for learning has taken a great importance in the whole teaching profession where it is one of the standards that a newly qualified teacher needs to meet. Hence it has become part of the professional skills and knowledge QTS standards (Q11, Q12, Q13, Q26, Q27, and Q28).

Due to the importance for Assessment for learning in teaching, the National strategies have developed with set of guidelines for assessing pupils' progress in learning which is called the APP guidelines. It is a structured approach to periodic assessment; enabling teachers to track pupils progress over a key stage or longer. It also allows using diagnostic information about pupils' strength and weaknesses to improve teaching and learning. Therefore, by using APP materials, teachers can make more consistent level related judgements in National Curriculum.

The APP for mathematics focuses on how a mathematics teacher can use AFL (Assessment for learning) strategy in lessons in order to generate evidence for pupil's learning. I have referred to the diagram below, to understand how the APP cycle works.

From the cycle, I am able deduce that assessing pupils' learning is not a one off task, it has to be on going and the teacher need to reflect on the evidence gathered through the assessment to adjust future planning for teaching.

By referring to the cycle, on a day to day teaching, a teacher could use strategies like probing questions, usage of mini white boards to assess pupils learning during the lesson. And of course going back to the traditional way of marking books will enable for the teacher to collect evidence for the pupils' learning. During my school placement, it was required by all staff in maths department to comment on the level of work each pupil has achieved for individual topic in the books that were being marked. Hence, by doing so the teachers were able review a range of evidence for a term of the pupils' progress in maths.

The assessment criteria sets out the different levels for each topic and clearly states what pupils are expected to know and able to do for each level. However, during my training in my first placement; I have encountered that the website called Kangaroo Maths - www.kangaroomaths.com, has an APP page that provides supporting materials for teachers from key stage 1 to 3. The assessment policy from the website (Appendix 1) has been rewritten to reflect the APP guidelines that help with the on going development of APP. It has an evaluation tool (appendix 2) where it allows teachers to self evaluate themselves in focusing, developing and establishing APP criteria with regards to pupils' engagement, lesson planning and evidence gathering. Further more, to understand the assessment criteria on the A3 grid, Kangaroo maths has developed the levelopaedias that provide exemplifications and probing questions for each of the assessment criteria. Therefore, by using these level ladders I have found easier to match the level of work done by the pupils where it gives me a guidance to plan my lesson accordingly and to gather the evidence on the level work achieved by the pupils.

Probing questions are an important tool in a lesson plan as it could be used to confirm pupils' understanding in a particular topic or their misconceptions. However, during my first placement, I have realised that having a set of suitable probing questions to assess learning for a particular topic was challenging because the questions you ask should be understood by the pupils before you could obtain the relevant answers for assessing learning. Hence, the structure of the questions is essential. In the APP criteria materials, there are some probing questions for each topic and level for teachers to use them to initiate dialogue as to assist in their assessment judgement. An example of a set of guidelines for adding and subtracting fractions has been show below;

Add and subtract fractions by writing them with a common denominator, calculate fractions of quantities (fraction answers); multiply and divide an integer by a fraction

Hence, by looking at the table above it is clearly stated what are pupils should know and what are the questions to ask assess their learning. As a trainee teacher, I think using these guidelines allows planning the lesson to promote effective learning and teaching.

DIFFERENTIATING

To promote effective learning and teaching of mathematics, it is important to differentiate your lesson according to the pupils learning styles or needs. Hence, the lesson plan should have included pupils who have learning disabilities and pupils who are Gifted and Talented; to ensure that equal opportunities for learning are given to all pupils in the class that leads to effective teaching.

DISCUSSION/FINDINGS:

To add on to my findings, I am going to look into the topic Algebra and analyse how it has developed across the levels using the APP criteria and Kangaroo maths Level Ladders. Then, based on level 5 work on Algebra, I am going to design 3 series of lesson plans with the guidance of the level ladders.

The word 'ALGEBRA' seems to be a put off to most students when unknown numbers or using formulas to real life context. It is a topic that requires accumulative understanding building on from level 2 onwards as shown below (taken from APP guidelines);

Algebra

Level 5 Construct, express in symbolic form and use simple formulae involving one or two operations.

Level 4 Begin to use simple formulae expressed in words

Recognise a wider range of sequences

Level 3 Begin to understand the role of '='

Level 2 To recognise sequences of numbers, including odd and even numbers

Design of Lesson plans:

I have designed the following lessons on Algebra based on a Yr 7 (lower sets) class working towards level 5 of work. I have used Collins New maths Framework, year 7 text book as basis for scheme of work.

Plan 1 (Appendix 2) - Calculating Using Rules

I have planned this lesson building on from level 3 to level 4 works; where as starter questions on sequences to revise level 3 work that the class should already know. I have used basic sequence worksheet (Appendix 8) from Kangaroo maths as it enables pupils to write down the following terms in the sequence and at the same time to identify the rule used for the sequence.

From the starter, I could introduce of using rules in calculation. As for the main part of the lesson, I am going to use an example where applying a rule to calculate the cost of a Window cleaner. I am going to ask the class as a probe, 'How much does it cost to clean per window?' So that I would be able make a judgement whether the class do understand the rule given charge= no. Of window cleaned x 40p. Then I will extend the question, 'what if the cleaner cleaned only 5 windows?'

As for plenary, I am going to use the example question again but with the rule, charge= no. of windows cleaned x 50p, calculate the cost for 8 and 5 windows cleaned. Again, to ensure that pupils do understand the rule and apply accordingly.

Plan 2 (Appendix 3) - Simplifying Expressions

The 2nd lesson I have planned based on Simplifying expressions at level 4. For this lesson, the class is expected to know about algebraic terms and expressions. Hence, I am going to have a starter where pupils are required to write terms and expression for given illustrations on to mini white boards. The questions will be flashed on power point slides.

However, I will have a quick revision on the terms and expression by asking a pupil to explain what 3N means? And N/2 means? In any case, if there were pupils do not understand the concept of 3N or N/2, I might lead them to finding unknown numbers, for example, 3x?=12..

As for the main part, I am going to play a video clip on introducing formulae from the following website;

http://www.bbc.co.uk/schools/ks3bitesize/maths/algebra/formulae1/activity.shtml

For plenary, I am going to go through the adding of like term questions. For example, write down the expression for 2cups add to 3 cups..2C+3C=5C. I will probing the class to explain what does 2C means? Again to judge whether the class understood the terms used.

Plan 3 (Appendix 4) - Solving Equations

For the third lesson, my lesson objective will be about solving equations. As for the starter, I am going to use the resources on Bring it on maths from Kangaroo maths on Simplifying Expressions (Appendix 8).

I am going to use unknown numbers to warm up the class to solve equations. For example, N + 5 = 9, I would probe the class,' how do you find N?'

As for plenary, I am going to create an algebra grid on a power point slide as follow;

Ethical Considerations:

In executing the lesson plans stated above, I need to look in to certain issues like differentiating my work at different level. Providing with Extension work for the G&T students; probing questions for the weaker ones to think and attempt questions. A separate set of work for LSAs supporting SEN/EAL students. Based on my observation on EAL students at Chase High School, I have learnt that EAL students tend to use visuals more often to recognise words used and described in maths. Hence, in my power points I have to add more visuals to get the EAL students to be engaged as well.